Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,47)(38,53)(40,48)(41,49)(42,56)(43,51)(44,52)(46,55)(50,54);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,31,40)(26,32,35,47)(28,39,30,37)(29,42,44,38)(33,48,45,36)(34,50,52,46)(41,54,43,55)(49,53,51,56);(2,5)(6,8)(11,17)(12,19)(14,20)(15,22)(25,28,31,30)(26,33,35,45)(27,37,40,39)(29,43,44,41)(32,36,47,48)(34,51,52,49)(38,55,42,54)(46,56,50,53);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,31,44)(26,34,35,52)(27,38,40,42)(28,41,30,43)(32,46,47,50)(33,49,45,51)(36,53,48,56)(37,54,39,55);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(29,44)(34,52)(38,42)(41,43)(46,50)(49,51)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436121);